What Goes Into Risk Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences

Transcription

1 Version of July 20, 2016 What Goes Into Risk Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences Stephen Figlewski* * Professor of Finance New York University Stern School of Business 44 West 4th Street New York, NY sfiglews@stern.nyu.edu tel: I thank the Q-Group and the NASDAQ OMX Derivatives Research Project at NYU for research support for this project. Thanks to Rob Capellini for preparing the GARCH forecasts. Valuable comments from Joanne Hill, Larry Harris, and participants at the Q Group 2012 Spring Seminar, the University of Michigan finance seminar, and the ITAM Finance Conference are greatly appreciated.

2 ABSTRACT What Goes Into Risk Neutral Volatility? Empirical Estimates of Risk and Subjective Risk Preferences Under Black-Scholes (BS) assumptions, empirical volatility and risk neutral volatility are given by a single parameter, which captures all aspects of risk. Inverting the model to extract implied volatility from an option's market price gives the market's forecast of future empirical volatility. But real world returns are not lognormal, volatility is stochastic, and arbitrage is limited, so option prices embed both the market's estimate of the empirical returns distribution and also investors' risk attitudes, including possibly distinct preferences over different volatilityrelated aspects of the returns process, such as tail risk. All of these influences are reflected in the risk neutral density (RND), which can be extracted from option prices without requiring restrictive assumptions from a pricing model. We compute daily RNDs for the S&P 500 index over 15 years and find that risk neutral volatility is strongly influenced both by investors' projections of future realized volatility and also by the risk neutralization process. Several significant variables are connected in different ways to realized volatility, such as the daily trading range and tail risk; others reflect risk attitudes, such as the level of investor confidence and the size of recent volatility forecast errors. Keywords: risk neutral volatility; implied volatility; option pricing; risk aversion JEL Classification: G13, G12, G14 2

3 The Black-Scholes (BS) model showed that option value depends heavily on the volatility of the underlying stock, which is assumed to follow a constant volatility logarithmic diffusion. With this process, a single parameter captures "risk" in all its manifestations. These include instantaneous volatility, which determines delta hedging performance; the standard deviation of the stock price around its mean on option expiration day, which is what implied volatility extracted from a pricing model measures; the probable trading range for the stock price over the option's lifetime, which is what determines whether a speculator can make a profit on a short-term trade and how much financing risk an arbitrage trade is exposed to; the frequency and size of tail events, which risk managers worry about most; and more. Different investors care about some of these risks more than others. For example, a market maker hedging his trading book worries about very short run fluctuations and price jumps that are of little consequence to a buy-and-hold investor. But under Black-Scholes, they all care about the same volatility parameter. Volatility in the BS model it is a known parameter, but real world volatility varies randomly over time and is hard to predict accurately. And no matter how volatility is estimated, market prices invariably differ from model values computed with that volatility input. As an alternative to statistical estimation, market makers and many others rely on implied volatilities extracted from option market prices. Implied volatility (IV) makes a theoretical model consistent with pricing in the market, but one has to specify which model "the market" is using. Black-Scholes IV is nearly always chosen, even though an enormous amount of research has shown that a constant volatility lognormal diffusion for returns is greatly oversimplified. 1 Constraining IV to be equal across strike prices, as the BS model requires, fails immediately in the face of the ubiquitous volatility "smile" (or "smirk" or "skew," depending on its shape). Equity market volatility changes over time; returns 1 See Poon and Granger (2003) or Jackwerth (2004) for reviews. 1

4 are not really lognormal, especially in the tails; and there appear to be non-diffusive jumps in both returns and volatility. Time-variation in the returns distribution should matter to investors, beyond the obviously greater difficulty of hitting a moving target, because stochastic volatility is unspanned (meaning it can't be fully hedged by other securities). Once more complex returns processes are allowed, numerous possibilities arise for unspanned stochastic factors that are not in the BS model but that can affect actual options and may be priced in the market. The resulting risk premia may themselves vary stochastically over time. Implied volatility is a "risk neutral" value, in which all risk premia in the market have been impounded into a modified probability distribution for returns, known as the Risk Neutral Density (RND). In an important paper, Harrison and Kreps (1979) proved that in a world free of profitable arbitrage opportunities, there will always exist a risk neutral density that combines investors' risk preferences with their objective forecast of the probability density for the stock price at option expiration. The RND is often called the "Q measure" while the expected true probability distribution is the "P measure." Implied volatility and other parameters computed under the Q measure will rarely be equal to their real world expected values under the P measure. Violations of the BS assumptions, like fatter than lognormal tails, cause BS implied volatilities to exhibit the customary but theoretically anomalous smile shape. Research on volatility risk, e.g., Carr and Wu (2008), shows that the market places a negative risk premium on volatility risk. The market dislikes volatility, so expected returns are lower and prices are bid up for options and other securities that can hedge volatility risk because they rise in price with higher volatility. They will be priced (under the Q measure) as if volatility were expected to be higher than investors really think. 2

5 IVs that differ across strikes is inconsistent with BS model assumptions. But when a crosssection of option prices is available, the full risk neutral probability density (RND) can be extracted without specifying a pricing model. Breeden and Litzenberger (1978) showed how this could be done very simply, given prices for options whose exercise prices span a broad range of future stock prices. Option prices are now being used to extract not just Black-Scholes implied volatilities, but the entire risk neutral probability density for the price of the underlying on expiration day. 2 The RND reflects current state of supply and demand for options with the same maturity and different exercise prices. It will impound the market's beliefs about the true probabilities of all volatilityrelated properties of the underlying stock's returns, such as time-variation in instantaneous volatility, fat tails and jumps, and it will also reflect risk premia on those factors and any others investors care about. Risk neutral volatility is the standard deviation under this density. We analyze these densities to explore how the RND, and by extension option pricing in the market, is affected by various volatility-related properties of the true returns distribution and by factors that may reflect risk preferences. In so doing, we begin to address the question in the title of this paper: "What is Risk Neutral Volatility?" The market bases its volatility forecasts and required risk premia on current and historical data. But these estimates apply to the behavior of the stock over the future lifetime of the options involved. Thus we are interested both in what information gleaned from historical data goes into the market's Risk Neutral Density, and also how that RND relates to the volatility that is actually realized in the future. We have divided the investigation into two parts. The current paper will focus on what past and current information appear to enter today's volatility estimates and risk preferences. A second paper will consider risk neutral volatility as a forecast, addressing such 2 See, for example, Bliss and Panigirtzoglou (2002), Giamouridis and Skiadopoulos (2012), Kozhan, et al. (2010), or Christoffersen, et al. (2011). 3

6 issues as: How accurate is it? How does it compare with other estimators, such as GARCH and the VIX index? How does the market's forecasting horizon compare to the option's remaining life? In the current paper, we address several broad questions, including: Which return and volatility-related factors are most important to investors in forecasting the empirical probability density that is embedded in the RND? What factors influence the process of risk neutralization? Do the answers to these questions differ for long maturity vs. short maturity options, or in different market conditions? Since risk premia are significant components of total return, there is considerable value in understanding risk-neutralization just to be able to estimate how much extra return the market requires for bearing different sorts of risks, or the flip side of the same question, how much would the market pay to hedge those risks? For example, Bollerslev and Todorov (2011) argue that much of the volatility risk premium during the fall of 2008 was compensation for left tail "crash risk," while Kozhan et al. (2010) find that about half of the implied volatility "skew" reflects a risk premium for skewness (i.e., the 3rd moment of returns). We use the Breeden-Litzenberger technology to extract RNDs from daily S&P 500 index option prices. The primary objective is not to build a formal model of option risk premia, but to establish a set of empirical stylized facts that any such model will need to be consistent with. Two significant problems, that we have addressed in a technical Appendix, are how to smooth and interpolate market option prices to limit the effect of pricing noise, and how to extend the distribution into the tails beyond the range of traded strike prices. An enormous advantage of the RND is that it does not require us to know the market's option pricing model. It is model-free. The VIX index that measures implied volatility for the overall stock market portfolio is roughly based on the same model-free methodology that we employ. 3 However, the VIX focuses on a single 30-day maturity, while maturities in our sample range from 14 to 199 days. The VIX 3 See Chicago Board Options Exchange (2003). 4

7 calculation also ignores several issues of data handling that we try to deal with more carefully, including extending the tails of the density beyond the range of available strike prices. 4 Still, it is worth noting that over our 15-year sample period, the VIX averaged 22.1 percent while realized volatility was only 20.6 percent on average. This implies a volatility risk premium on index options equivalent to about 1.5 volatility points. Before moving on, we must consider how the recent theoretical breakthrough in this area by Ross (2015) applies to this research. It has long been thought that objective probability estimates and risk premia cannot be separately deduced from option prices without further information about either the market's risk preferences or the true returns generating process. Ross proves a deep theoretical result showing this is not strictly true. But the assumptions required are strong enough that the immediate problem we are addressing remains, which is to understand better how objective measures of different aspects of price risk and behavioral proxies for market "sentiment" get into real world option prices. The next section defines the variables we conjecture could influence investors' demands for options. A practical issue that is rarely considered is how far back in time investors look in calculating volatility and other statistics from past returns. In this research, we have investigated different lag definitions as well as different cutoffs to define tail events and assume investors pay most attention to those that contain the most information about future volatility. Due to space limitations, the Appendix is available on request to the author. Section 3 first looks at univariate relationships in the form of simple correlations between the volatility measures and the explanatory variables, and then presents a grand regression with all of the variables together. Among the interesting results from this exercise is strong evidence that several distinct aspects of volatility seem to be independently important, and variables that were 4 Jiang and Tian (2007) demonstrate that ignoring these features in the VIX methodology introduces substantial and unnecessary inaccuracy. 5

8 hypothesized to be correlated with investor risk attitudes do in fact appear to have significant explanatory power for RND volatility. Section 4 explores robustness of the model by fitting it on subsets of the data, according to option maturity and the time period, and the final section offers some concluding comments. Section 2: Data The variable we wish to explain is the standard deviation of the S&P 500 index on option expiration day under the risk neutral probability density. The RND aggregates the individual risk neutralized subjective probability beliefs within the investor population. The resulting density is not a simple transformation of the true but unobservable distribution of realized returns on the underlying asset as in Black-Scholes, nor should we expect it to obey any particular probability law. Our model-free fitting procedure allows maximum flexibility to reflect pricing in the options market. To compare RNDs from options with different maturities, the RND standard deviation is converted into an annualized percentage value. This is what we call Risk Neutral Volatility. The basic strategy of non-parametric RND extraction is well-known by now, so to save space the specifics of our implementation are sketched out in the Appendix that provides more details about this and other technical aspects of the empirical work. 5 RND and Realized Volatility Option prices: We take the midpoints of the closing bid and ask quotes for all traded S&P 500 index options, downloaded from OptionMetrics, that satisfied the following criteria: The observation date and option expiration were between Jan. 4, 1996 and Apr. 29, Option Bid price > The average index level in the sample is See also the Appendix to Birru and Figlewski (2012) for full detail on the RND extraction methodology. 6

9 Maturities 14 < T < 199 calendar days, allowing about three RND maturities per day. Interest rates and dividend yield: Riskless interest rates, interpolated to match the option maturity, and S&P 500 dividend yields were obtained from OptionMetrics. Realized volatility to expiration: An option's price should embed the market's best forecast of the volatility that will be realized from the present through option expiration, but this is not the volatility under the RND when there is a volatility risk premium. We explore the real world differences, by regressing RND volatility and future realized volatility on explanatory variables that we hypothesize are important to investors. Realized volatility for date T maturity is defined as (1) This assumes 252 trading days in a year, and adopts the standard approach of treating the daily mean return as equal to 0. 6 Realized volatility is expressed in annualized percent, e.g., 20% is We consider explanatory variables in three sets. First are variables related to volatility and extreme price movement, that do not require analyzing historical data. In the second set are volatility-related variables calculated from historical data. The third set are variables chosen to reflect behavioral factors, that we feel might affect the risk neutralization process. Date t Volatility Variables Date t return: The Date t return is defined as 100 x log(st / St-1). Volatility, both empirical and implied, has a strong tendency to go up following a negative return. Models with separate stochastic factors for returns and volatility typically find correlation of about -0.7 between them. Absolute value of date t return: A large return shock of either sign is consistent with high current volatility, but this variable was replaced with a different but related one, as described below. 6 The average daily return is only a few basis points, which is much smaller than the average sampling error on the mean, so it is generally felt that the small bias introduced by treating the mean as zero is preferable to the much larger errors that would result from using the sample mean in the calculation. 7

10 specification: GARCH model forecast: We adopt the Glosten, Jagannathan, and Runkle (1993),, ~ 0,1 1 (2) where rt and t 2 are the date t log return and conditional variance, and zt is a standard normal i.i.d. random variable. The return equation contains no constant term, consistent with eq. (1). The variance t 2 is a function of the date t-1 conditional variance and squared return shock, plus a term that increases t 2 when rt-1 is negative. GJR-GARCH model forecasts were provided by the NYU Volatility Institute's VLAB. 7 Date t trading range: The intraday trading range contains a lot of volatility information. 8 Also the cost for market makers to hedge their "Greek letter risks" depends on the range over which the stock price travels in a day. For consistency with the other variable definitions, the Date t range is converted into a logarithmic percentage in the following way: Range(t) = 100 x log(1 + (St,HI - St,LOW ) / ((St,HI + St,LOW ) /2))) (3) Like the absolute return for the day, this variable was included in our initial exploration, but the results seemed odd. In regressions with both the absolute return and the trading range, trading range comes in with a highly significant positive coefficient but the coefficient on absolute return consistently was about the same size but strongly negative. Subsequently, market makers explained that it is much easier for them to manage risk on a day when the market simply moves strongly up 7 A GJR-GARCH model was fitted for every day during the sample and out of sample variance forecasts were generated for 1 to 252 trading days ahead. These were used to construct predictions of average variance from each observation date through option expiration. The resulting variances were converted to annualized percent volatilities. See 8 Parkinson (1980) showed that a volatility estimator based on the intraday high and low can achieve the same accuracy as one using only close to close returns with five times as many observations. 8

11 or down, than when there is high uncertainty and prices fluctuate over a broad range without finding a new equilibrium much different from the initial level. They suggested using instead: Date t trading range minus absolute return: The range minus the absolute change produced a single variable with a strongly positive and significant coefficient. Date t left and right tail returns: One of the most apparent differences between the empirical returns distribution and the lognormal is that large absolute returns tail events are more common than the lognormal allows for, especially on the downside, so the occurrence and size of a tail event might be especially important to investors. But what constitutes a tail event? Rather than making ad hoc assumptions, as is usually done, I treated as an empirical question, which was explored by comparing the explanatory power of alternative tail definitions, both in univariate regressions of RND volatility on a tail variable, and also along with our other explanatory variables in multivariate specifications. Based on this investigation, as detailed in the technical Appendix, I classify the return on date t as a left (right) tail event if it falls in the 2% left (right) tail (relative to a normal density). The cutoffs for tail returns are set at ±2.054 times the previous day's 1-day GARCH volatility forecast. If the return is not in the tail, the value of the tail variable is set to 0. Historical Volatility Variables Past realized volatility and similar factors are computed from historical returns, but it is not known a priori what sampling horizon the market focuses on. As with tail definitions, I used the behavior of the RND as a window onto this issue. Most of the variables showed similar performance over a fairly broad range of sample lengths. In the end, I selected estimation windows that gave reasonable performance in the various exploratory regressions and that "made sense" in terms of the behavior one might expect from an intelligent investor. Details are given in the Appendix. The final choices are as follows. 9

12 Left and right tail events: The empirical distribution's fat tails can be due to more frequent tail returns than under the Normal and/or unexpectedly large tail outcomes conditional on falling in the tail. As mentioned above, 2% and 98% cutoffs define a tail event. A historical window of 2 years (500 trading days) was chosen to gauge frequency for both tails. Return and Absolute return from date t-n to t-1: Recent index return is defined as 100 x log(st-1 / St-n), but the "right" choice of n was not clearcut. The return over just the last 5 trading days was selected because it had good explanatory power and less chance that it is proxying for some other factor. Historical volatility: Historical volatility is computed from the squared log returns over the last 65 trading days, treating the expected drift as zero. Specifically, (4) Average daily range t-n to t-1: The daily range is a proxy for potential hedging costs. Each day's range is computed as in eq. (3). In the technical Appendix we find that averaging the daily range over a window the same size as the option's remaining lifetime performs distinctly better than any fixed horizon. Range over t-n to t-1: If the underlying stock trades over a wide range during an option's lifetime, an investor may make a short term profit on an option early even if it ultimately expires out of the money. The range the index has traversed in the most recent 25 days is computed as a logarithmic percentage relative to the average price in the interval. GARCH error and GARCH RMSE: Investors who dislike volatility and also volatility uncertainty may require a larger risk premium when their forecasting model has been performing badly, so we incorporate two variables measuring historical accuracy of the GARCH model. A positive coefficient on the average GARCH error from date t-n to t-1 means that RND volatility is higher when the GARCH model has been underpredicting realized variance, while a positive 10

13 GARCH RMSE coefficient increases risk neutral volatility following large model errors on either side. The historical sample length over which investors are assumed to gauge the GARCH model's accuracy is set equal to the option's remaining lifetime. Risk Attitude Variables The variables discussed so far are all tied to statistical characteristics of returns. We now turn to the risk neutralization process. I have selected a few factors that are potentially promising, but there are many others that it might make sense to explore. University of Michigan Index of Consumer Sentiment: This is a classic survey of consumers' confidence level. It is not specifically tied to finance, but it is widely known and followed by investors. The daily value is set to the monthly sentiment index value for every day within the month. Baker-Wurgler Index of Investor Sentiment: Baker and Wurgler (2006) devised a measure of investor confidence based on factors such as the discount on closed-end mutual funds, the first day returns on IPOs, etc. With all monthly series, the value is repeated for every day in the month. The Price/Earnings Ratio for the S&P 500 index portfolio: Price/earnings ratios will be higher when investors anticipate favorable conditions and strong earnings growth in the future. We hypothesize that optimistic investors also accept lower volatility risk premia. The monthly P/E ratio for the S&P index was downloaded from Robert Shiller's "Irrational Exuberance" website. 9 The Baa-Aaa Corporate Bond Yield Spread: This daily credit spread series is the average yield spread between corporate bonds rated Baa by Moody's and AAA bonds. It measures the default risk premium on high grade corporate debt. The series was downloaded from the St. Louis Fed's FRED database

14 GARCH model error over a recent period equal to the option's remaining lifespan: A positive (negative) value corresponds to average under- (over-) prediction of the squared returns. If GARCH model errors in previous periods affect current risk neutral volatility, it is more likely through risk aversion than through the objective conditional forecast of future volatility. GARCH Model RMSE: This variable is described above, and in the technical Appendix. RND Volatility Premium on Date t-1: If "market sentiment" is persistent, perhaps due to excluded exogenous variables, that dependence can be captured by including the previous day's volatility risk premium, which we measure as date t-1 RND volatility minus the GARCH forecast of volatility to expiration. But simply "explaining" risk neutral volatility by its own lagged value is not very revealing, so this variable is not included in the "all variables" specification below. It is useful, however, to see if yesterday's volatility risk premium contains information for predicting realized volatility, and also to see how the coefficients on the other variables are affected if this variable is included in the regression. Section 3. Results We first look at univariate correlations among the volatility-related dependent variables and the explanatory variables. This will show the typical explanatory power of each factor when considered in isolation. We then fit a grand regression on all variables together to gauge each one's marginal contribution when included with the others. Because RND volatility impounds a forecast of volatility over an option's remaining lifetime, there is potential correlation in residuals from any two options whose lifetimes overlap. This does not bias coefficient point estimates, but OLS standard errors are inconsistent. The standard errors are corrected using an adaptation of the Newey and West (1987) model, as detailed in the technical Appendix. Simple Correlations 12

15 Table 1 shows simple correlations between the dependent variables and each of the explanatory variables. With more than 20 of them, multicollinearity is a concern, but only a few correlations are above 0.7 and these tend to be where one would expect them. For example, variables like historical volatility and the daily trading range are highly correlated with the GARCH forecast, as one might expect. Yet, these correlations are not so high that they preclude precise coefficient estimates. 11 Recall that in a one-variable regression, the correlation and regression coefficient have the same sign, and the R 2 is the square of the correlation coefficient. We first note that the correlation between RND volatility and Realized volatility to expiration is only The low value reflects the fact that risk neutralization modifies the market's prediction of the empirical density and, moreover, future volatility is just hard to forecast accurately, e.g., the GARCH forecast and historical volatility show similar correlation levels with Realized volatility. The next section contains the Date t variables. Date t return is negatively correlated with the two dependent variables, but only weakly, probably because the table reports correlations among levels, not changes. The absolute value of Date t return is distinctly more important than the signed return, especially for RND volatility. The GARCH forecast of volatility through expiration is an extremely important variable. Its correlation with both RND and Realized volatility is close to the largest for any of the exogenous variables. 12 Extreme negative and positive "tail" returns show the expected signs--a large negative left tail return leads to higher volatility and so does a positive return in the 2% right tail--but correlations are low. Next are the volatility-related variables computed from historical prices and returns. With negative correlations on last week's return and positive correlations on the absolute return, if the 11 The full correlation matrix for all dependent and exogenous variables is shown in the Appendix. 12 Although the model is fitted on historical data, GARCH is a "Date t" variable because ambiguity over how far back the historical data sample should go is not a major issue with a parametric model like GARCH. 13

16 market went up (down) over the past week, volatilities fell (increased), but volatility was higher after a week with a large return of either sign. Historical volatility should be similar to GARCH but less accurate as a forecast since it uses the same past returns, but in a less sophisticated way. Still, investors may well incorporate it in RND volatility because it is more accessible to many market participants than a full-fledged GARCH model. As expected, the correlations between Historical volatility over the last 65 days and the two dependent variables are very high, but for Realized volatility a little lower than the GARCH forecast. Looking backward, two different trading range concepts could be important. If the index moves over a broad range intraday it signals high volatility and also greater hedging risk for market makers. The daily range averaged over a longer time period measures the latter effect in recent returns, and the correlations with the two volatility variables are strongly positive, more so for risk neutral volatility. 13 The second trading range concept is of more interest to a longer term investor. Trading over a wide range during the option's lifetime can be desirable if it gains the investor the opportunity for a short run profit even when the option in question will eventually expire out of the money. While RND volatility is very highly correlated with both range variables, they have less explanatory power for the volatility that is actually realized. The next four variables relate to the frequency and size of past extreme events. Seeing many returns in the tails should increase investors' perceptions of volatility risk, and the table shows positive correlations with the number of both left and right 2% tail events. The average size of a tail return is another dimension of tail risk. The average left tail return is negative, so negative correlation means volatility increases when a big negative left tail return happens. The right tail return shows the expected positive correlation. Realized volatility is also related to these tail variables but with substantially lower correlations than RND volatility shows, in each case. 13 Based on the analysis in the technical Appendix the option's remaining lifetime is selected as the horizon. 14

17 The last section of Table 1 contains variables related to the risk neutralization process. Both RND and Realized volatility are negatively correlated with the University of Michigan Survey of Consumer Sentiment and the P/E ratio for the S&P index. The Baker-Wurgler measure, however, is less correlated with RND volatility and has the "wrong" sign for Realized volatility, although the size is quite small. To the extent that these variables reflect investor confidence, negative correlation with RND volatility is expected. The Baa - Aaa corporate bond yield spread measures market-perceived default risk. The variable is strongly positively correlated with the two volatility variables, which suggests that wider credit spreads in the bond market correspond to higher volatility in equity options. The next two variables capture how accurate the GARCH model forecast has been in the recent past. Investors perceive more volatility risk, and there seems to be more actual volatility, following a period in which volatility models exhibited large errors, especially if volatility was underpredicted during the sample period (i.e., mean GARCH error was positive). Both RND and future Realized volatility are strongly correlated with GARCH RMSE. The final variable in the table is the previous day's RND volatility premium. There is almost no correlation between yesterday's volatility risk premium and today's RND volatility level (although the premium itself is highly autocorrelated), and the correlation with Realized volatility is actually negative. Thus the difference between risk neutral volatility and a good forecast of empirical volatility does not appear to contain information about the volatility that will be realized in the future or even about tomorrow's risk neutral volatility. The correlations in Table 1 show that the factors that were expected to influence RND volatility and Realized volatility appear to do so with the anticipated signs. But correlation among the explanatory variables is high, so it is not clear how much any individual factor contributes 15

18 beyond the fact that all of them are correlated with "volatility" broadly defined. To gauge the marginal influence of each variable, we now combine them all into a single grand regression. Regressions with All Variables Table 2 presents the results from four regressions run with the full set of explanatory variables, with t-statistics corrected for cross-correlation. The first four columns feature key results and some interesting differences between RND volatility and Realized volatility. Both show a significant "leverage effect", with negative Date t returns leading to higher volatility, and highly significant coefficients on the GARCH forecast. However, the coefficient on GARCH is more than twice as large for Realized volatility as for RND volatility and both are well below 1.0. Trading on Date t over a much wider range than the change from yesterday's close is associated with higher risk neutral volatility and even more strongly with subsequent realized volatility. But left or right tail returns had no significant effect on subsequent realized volatility. The t-statistics in the RND volatility regression are much higher, but the anomalous positive coefficient on a negative tail return implies that a shock to the downside reduced RND volatility. For RND volatility, all nine Historical returns variables except the average left tail return came in significant with the expected signs, while only three had significant explanatory power for Realized volatility. A behavioral explanation is that different volatility-related aspects of the returns process may be independently relevant to, and valued by, certain classes of investors beyond their connection to future realized volatility. Market makers may focus especially on the recent daily range, while speculators may favor options on stocks that have traded over a wide range in the past month, and the most risk averse investors might be particularly worried about tail events. Including Historical volatility and GARCH in the same specification allows for the expected result that GARCH, as a superior forecaster, should dominate Historical volatility in the Realized volatility equation. Indeed, Historical volatility gets a negative coefficient in the Realized volatility 16

19 equation, even though the simple correlation between the two is Yet Historical volatility could be important to investors who are less comfortable with sophisticated econometrics. And, in fact, both GARCH and Historical volatility do get significantly positive coefficients in the RND equation, albeit with the GARCH coefficient being more than three times as large. Among the risk neutralization variables, strength in the Michigan Survey, the Baker- Wurgler investor confidence measure and the market price/earnings ratio were associated with lower RND volatility, although only the Michigan Survey was significant. The Michigan sentiment measure was also significant for Realized volatility, but Baker-Wurgler investor sentiment came out strongly significant with the wrong sign. The Baa-Aaa bond yield spread was not significant for RND volatility, but it had the expected positive sign and was quite large and nearly significant for Realized volatility. In contrast to Table 1, the two variables related to accuracy of the GARCH model were insignificant, and 3 of 4 had the wrong signs. The four rightmost columns present results from the same regressions with yesterday's volatility premium added as an explanatory variable. We do not favor this specification for the RND volatility regression because, while yesterday's volatility premium is obviously highly significant, this difference is a large part of what we are trying to explain with exogenous factors. We include it here simply to show that it has little effect on the other coefficients. There were changes in coefficient sizes and significance levels, but just a few variables changed sign and only the Date t right tail return went from significantly positive to significantly negative. The lagged RND volatility premium in the Realized volatility regression is nearly significant but negative, and it contributes almost nothing to raising the regression R 2. This variable reflects the risk neutralization process, which is not a prediction of future volatility at all. A negative coefficient suggests that holding the other predictive variables constant, future realized volatility tends to be lower (higher) when the volatility risk premium is high (small). 17

20 Section 4: Subsample Results This section explores the robustness of the relationships uncovered in the last section in subsamples broken down by maturity and in different economic environments. Table 3 runs the regression of RND volatility on the full set of explanatory variables separately for contracts with less than 75 days to expiration and for longer maturity contracts. The results for short term and longer term options are remarkably similar. Only the average size of left tail events changes sign, and is insignificant in both subsamples. The average right tail return in the last two years becomes significant in the longer maturity regression, while the coefficient on GARCH RMSE becomes insignificant (with the wrong sign in both regressions). Table 4 compares regressions for both RND volatility and Realized volatility on all variables over three different time periods. The first is January June 2003, which included the Russian debt crisis and the Internet bubble, followed by the bear market of It was a period of medium volatility with a sharp run-up in stock prices and a sharp drop in the second half. The second subsample, July December 2007, was a time of persistently rising stock prices and extraordinarily low volatility. The third subperiod, January April 2011 includes the market crash in 2008, when volatility soared to extreme heights, followed by the aftermath. Not surprisingly, there is considerable dispersion but the important full sample results largely hold across subperiods for both RND and Realized volatility. The differences tend to be more in statistical significance levels than in the fitted coefficients. The GARCH forecast remains highly significant, except for Realized volatility in the second subperiod. Historical volatility is positive and significant in two of three RND volatility regressions, but insignificant in two subperiods for Realized volatility and significant but with a negative sign in the third. 18

21 The Date t and Last week return variable coefficients were negative in nearly all cases, and significant for most of them. These variables were consistently more significant in the RND volatility than the Realized volatility equations, suggesting that returns may affect risk attitudes in addition to objective volatility expectations. The Date t trading range minus the absolute return was positive in all six regressions and highly significant in five of them. A large average daily range in the past increased RND volatility significantly in every subperiod, but Realized volatility only in the earliest one. The range covered by the index over the previous 25 days did not perform consistently. The coefficients in the RND volatility regressions were positive, and significant overall in Table 2, but only significant in the final subperiod, while for Realized volatility this variable was significantly positive in the earliest subperiod and significantly negative in the last. The left and right tail variables did not perform well. A Date t left tail return received a significantly positive coefficient in all three RND volatility regressions, implying that an unusually large negative return was associated with a decrease in risk neutral volatility. Frequent left tail events in the past increased both risk neutral and realized volatility overall, but results varied widely across subperiods. Similarly, the average size of left tail events showed inconsistent coefficient estimates and more than half had the wrong (i.e., positive) sign. Right tail events showed a similar lack of consistency between RND volatility and Realized volatility equations and over time. Among the Risk neutralization variables, a higher value for the Michigan survey reduced both RND volatility and Realized volatility significantly, with a couple of exceptions, while the Baker-Wurgler measure of investor confidence was more ambiguous, with both volatility measures increasing when this confidence measure was higher in the middle subperiod. By contrast, the S&P 500 P/E ratio had the right (negative) signs in Table 2, but was not significant. Here, the coefficients are negative in all regressions, and half are significant. Overall, the coefficient on the 19

22 bond yield spread was insignificant for RND volatility but positive and significant for Realized volatility, while in Table 4, it is significantly positive for Realized volatility only in the final subperiod, consistent with default risk becoming a more important factor after Finally, the GARCH model errors and the GARCH RMSE continue to be largely insignificant and anomalous in the subsamples as they were in Table 2. A negative coefficient on GARCH RMSE implies that RND volatility is lower when the GARCH model has been less accurate in the recent past, and a negative coefficient on the GARCH error means that RND volatility was lower still when GARCH volatility was underpredicted. To summarize the results of these subsample regressions, there was not much difference between short and long maturity contracts in their sensitivities to the explanatory variables. More diverse results showed up when the sample was broken into subsamples by time period. High predicted volatilities from both the GARCH and historical volatility models were associated with significantly higher risk neutral and future realized volatilities in most cases, although the relationship was weaker for historical volatility in the Realized volatility regressions. The return variables showed the expected strongly negative correlation between return and volatility, while absolute return was more ambiguous. The variables relating to the trading range intraday and over a longer historical period were for the most part strongly positive: trading over a wider range increased RND volatility, and Realized volatility too, in most cases. The tail event frequency and size showed inconsistent results, with some coefficients significant, but with possibly different signs in different subperiods, and quite a few anomalous values. The results on proxies for the market's risk tolerance were promising, though not overpoweringly strong. In most cases, positive sentiment seems to be associated with lower volatility. However, the Baa-Aaa bond yield spread and the two variables measuring the accuracy of the GARCH model did not add much explanatory power. 20

23 Section 5: Concluding Comments We set out to develop stylized facts about what factors determine risk neutral volatility. Let us summarize what we have uncovered. The first question was what return and volatility-related factors are most important to investors. Under Black-Scholes this amounts to getting the most accurate estimate of the instantaneous volatility. We found that RND volatility is strongly, but not perfectly, correlated with Realized volatility, with correlation around 0.6. Forecasts of realized volatility from a GARCH model are highly significant in explaining RND volatility. But investors seem to care about other features of the returns process beyond expected future volatility like the recent trading range, and the risk premia they require appear to be related to measures of confidence and sentiment. These non-black-scholes factors are important determinants of how options are priced in the real world. A GARCH model forecast was found to be highly significant for both Realized and RND volatility. As a less accurate predictor, Historical volatility did not contribute significantly in forecasting Realized volatility but it did in the RND volatility regressions. Investors appear to pay attention to both estimators in pricing options. The well-known negative relationship of volatility with return was strongly manifested by risk neutral volatility. Both the Date t return and also the return over the previous week were estimated with highly significant negative coefficients overall and, with a single exception, in every subperiod regression. The larger and more significant coefficients in the RND volatility regressions suggest that empirical ("P measure") volatility is correlated with returns, but transforming the P measure into the Q measure increases the effect. Trading over a wide range that produced little change in the closing index level was highly significant in increasing both RND and Realized volatility. But a large average daily range in the recent past or a large total range also increased RND volatility substantially, while having no 21

24 significant effect on realized volatility. These variables seem to be more strongly connected to investors' and market makers' risk aversion than to their objective forecasts of future volatility. Six variables were specifically chosen for their possible correlation with risk preferences. RND volatility was negatively correlated with the Michigan survey of consumer sentiment and the Baker-Wurgler index of investor sentiment, with the former showing strong performance for both RND volatility and Realized volatility. It is not surprising to find consumers and investors less confident in times of high market volatility. The effect appears distinctly stronger for RND volatility than Realized volatility, supporting the hypothesis that investor sentiment impacts risk neutralization. The price/earnings ratio of the S&P 500 and the credit spread in the bond market are direct measures of the market's willingness to bear risk. The consistently negative coefficient on the P/E ratio suggests that it may also be a useful reflection of market sentiment, although it was only significant in a few cases. However, the bond yield spread results were odd overall and inconsistent in the breakdown across subperiods. Lastly, the two variables designed to measure investors' confidence in the GARCH model's accuracy did not add explanatory power. The coefficients were mostly insignificant with anomalous signs. 22

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